Multidimensional bent functions

Overview & Synopsis

The concept of bent functions, originally introduced by Dillon and Rothaus, is very relevant in cryptography because this kind of functions represents the maximal resistance against the so-called linear cryptanalysis. In 1997, Logachev, Salnikov and Yashchenko described a fundamental notion of bentness for functions defined on a finite Abelian group $G$ with values in the unit circle of the complex field. In this paper, by replacing this unit circle by the unit hypersphere $\mathcal{S}_{\mathcal{H}}(0_{\mathcal{H}},1)$ of an arbitrary finite-dimensional Hermitian space $\mathcal{H}$, we develop a generalization of the concept of bentness for $\mathcal{S}_{\mathcal{H}}(0_{\mathcal{H}},1)$-valued functions defined on $G$, called multidimensional bent functions.